Properties of Screw Dislocation Dynamics: Time Estimates on Boundary and Interior Collisions

Hudson, Thomas E.; Morandotti, Marco

doi:10.1137/17m1119974

Cited by 5 publications

(6 citation statements)

References 29 publications

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“…where we have applied (G). Setting w = v i in (47) and bounding the left-hand side below as in (45), we thereby obtain…”

Section: Evolution Problem and Existence Resultsmentioning

confidence: 99%

“…The fact that the dislocations alone are tracked in this approach has the advantage of drastically reducing the computational complexity in comparison with phase field approaches, and as such, DDD has been used as a simulation technique for studying plasticity since the early 1990s [5,6,8,17,33,44,77]. While a significant mathematical literature has developed which considers one-and two-dimensional DDD models for the motion of straight dislocations [1,2,12,13,15,23,24,46,47,78], few mathematical results concerning DDD in a three-dimensional setting exist to date, and in part, this appears to be due to the lack of a clear mathematical statement of what the evolution problem for DDD should be in this setting.…”

Section: Introductionmentioning

confidence: 99%

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An existence result for Discrete Dislocation Dynamics in three dimensions

Hudson¹

2018

Preprint

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We present a mathematical framework within which Discrete Dislocation Dynamics in three dimensions is well-posed. By considering smooth distributions of slip, we derive a regularised energy for curved dislocations, and rigorously derive the Peach-Koehler force on the dislocation network via an inner variation. We propose a dissipative evolution law which is cast as a generalised gradient flow, and using a discrete-in-time approximation scheme, existence and regularity results are obtained for the evolution, up until the first time at which an infinite density of dislocation lines forms.

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“…where we have applied (G). Setting w = v i in (47) and bounding the left-hand side below as in (45), we thereby obtain…”

Section: Evolution Problem and Existence Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An existence result for Discrete Dislocation Dynamics in three dimensions

Hudson¹

2018

Preprint

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“…The regularity of required in the introduction can be weakened, especially for the results concerning the confinement; the details can be found in the papers [7,8].…”

Section: Discussionmentioning

confidence: 99%

“…Theorem 1 (free boundaries attract dislocations [7,Theorem 3.1] Moreover, the following estimate on the collision time holds: .…”

Section: Boundary Behaviormentioning

confidence: 99%

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Boundary Behavior and Confinement of Screw Dislocations

Morandotti

2017

MRS Advances

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In this note we discuss two aspects of screw dislocations dynamics: their behavior near the boundary and a way to confine them inside the material. In the former case, we obtain analytical results on the estimates of collision times (one dislocation with the boundary and two dislocations with opposite Burgers vectors with each other); numerical evidence is also provided. In the latter, we obtain analytical results stating that, under imposing a certain type of boundary conditions, it is energetically favorable for dislocations to remain confined inside the domain.

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The continuum limit of interacting dislocations on multiple slip systems

Meurs

2020

ESAIM: COCV

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In this paper we derive the continuum limit of a multiple-species, interacting particle system by proving a Γ-convergence result on the interaction energy as the number of particles tends to infinity. As the leading application, we consider n edge dislocations in multiple slip systems. Since the interaction potential of dislocations has a logarithmic singularity at zero with a sign that depends on the orientation of the slip systems, the interaction energy is unbounded from below. To make the minimization problem of this energy meaningful, we follow the common approach to regularise the interaction potential over a length-scale δ > 0. The novelty of our result is that we leave the type of regularisation general, and that we consider the joint limit n to infinity and δ to 0. Our result shows that the limit behaviour of the interaction energy is not affected by the type of the regularisation used, but that it may depend on how fast the size (i.e., δ) decays as n to infinity.

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Properties of Screw Dislocation Dynamics: Time Estimates on Boundary and Interior Collisions

Cited by 5 publications

References 29 publications

An existence result for Discrete Dislocation Dynamics in three dimensions

An existence result for Discrete Dislocation Dynamics in three dimensions

Boundary Behavior and Confinement of Screw Dislocations

The continuum limit of interacting dislocations on multiple slip systems

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